Numerical solution of time-fractional fourth-order reaction-diffusion model arising in composite environments

Abstract

The fractional reaction-diffusion equation has an important physical and theoretical meaning, but its analytical solution poses considerable problems. This paper develops an efficient numerical process, the local radial basis function generated by the finite difference (named LRBF-FD) method, for finding the approximation solution of the time-fractional fourth-order reaction-diffusion equation in the sense of the Riemann-Liouville derivative. The time fractional derivative is approximated using the second-order accurate formulation, while the spatial terms are discretized by means of the LRB-FFD method. The advantage of the local collocation method is to approximate the differential operators by a weighted sum of the values of the function on a local set of nodes (local support) by deriving the RBF expansion. Furthermore, the ill-conditioned matrix resulting from using the global collocation method is avoided. The unconditional stability property and convergence analysis of the time-discrete approach are thoroughly proven and verified numerically. Three numerical examples are presented confirming the theoretical formulation and effectiveness of the new approach.